What is the derivative of a constant?

Zero. The derivative of any constant is 0, because a constant has no rate of change. For example, d/dx(7) = 0.

What is the derivative of sin(x)?

The derivative of sin(x) is cos(x). The derivative of cos(x) is -sin(x) — note the negative sign.

How do you find the second derivative?

Differentiate the function once to get f′(x), then differentiate again to get f″(x). The second derivative measures concavity and is used in inflection point tests.

What is the difference between a derivative and an integral?

A derivative measures instantaneous rate of change. An integral accumulates total area under a curve. They are inverse operations connected by the Fundamental Theorem of Calculus.

What does the derivative tell you about a graph?

The derivative at a point equals the slope of the tangent line. Positive derivative means the function is increasing; negative means decreasing; zero may indicate a local maximum or minimum.

Derivative Calculator

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What Is a Derivative?

A derivative measures the instantaneous rate of change of a function — the slope of the tangent line at any given point on a curve. If you know how a quantity is changing at every moment, you know its derivative.

Concrete example: If f(x) = x², then f′(x) = 2x. At the point x = 3, the slope of the curve is 2(3) = 6. The tangent line at that point rises 6 units for every 1 unit of run.

Formally, the derivative is defined as the limit of the difference quotient:

f′(x) = lim(h→0) [ f(x + h) − f(x) ] / h

In practice, you don't use this limit every time — you apply derivative rules instead. The calculator above applies these rules automatically and shows every step.

Derivative Rules — Quick-Reference Table

Memorising these rules lets you differentiate any standard function quickly.

Rule	Formula	Example
Constant	d/dx(c) = 0	d/dx(7) = 0
Power	d/dx(xⁿ) = nxⁿ⁻¹	d/dx(x⁵) = 5x⁴
Constant Multiple	d/dx(cf) = c · f′	d/dx(3x²) = 6x
Sum / Difference	(f ± g)′ = f′ ± g′	d/dx(x³ + x) = 3x² + 1
Product	(fg)′ = f′g + fg′	d/dx(x · sin x) = sin x + x cos x
Quotient	(f/g)′ = (f′g − fg′) / g²	d/dx(x/eˣ) = (eˣ − xeˣ) / e²ˣ
Chain	d/dx[f(g(x))] = f′(g(x)) · g′(x)	d/dx(sin(3x)) = cos(3x) · 3
Exponential (eˣ)	d/dx(eˣ) = eˣ	d/dx(e²ˣ) = 2e²ˣ (chain rule)
Natural Log	d/dx(ln x) = 1/x	d/dx(ln(5x)) = 1/x (chain rule)
Sine	d/dx(sin x) = cos x	d/dx(sin(x²)) = cos(x²) · 2x
Cosine	d/dx(cos x) = −sin x	d/dx(cos(3x)) = −sin(3x) · 3
Tangent	d/dx(tan x) = sec²x	d/dx(tan(2x)) = 2sec²(2x)

Worked Examples — Step by Step

Power Rule: f(x) = 3x⁴ − 5x + 2

Differentiate each term separately using the power rule, then combine:

Differentiate 3x⁴. Bring the exponent down and reduce it by 1: 4 × 3x³ = 12x³.
Differentiate −5x. The exponent on x is 1, so: 1 × (−5)x⁰ = −5.
Differentiate the constant 2. The derivative of any constant is 0.
Combine. f′(x) = 12x³ − 5.

Chain Rule: f(x) = sin(3x²)

The chain rule applies whenever you have a function inside another function. Identify the outer and inner functions first:

Identify the outer function. sin(u) — where u = 3x².
Differentiate the outer function. d/du(sin u) = cos u.
Differentiate the inner function. d/dx(3x²) = 6x.
Multiply outer by inner derivative (chain rule). f′(x) = cos(3x²) · 6x.
Simplify. f′(x) = 6x cos(3x²).

Why multiply? The chain rule says the rate of change of the composite function equals the rate of the outer function (evaluated at the inner) times the rate of the inner function. You're chaining the rates together.

Quotient Rule: f(x) = (x² + 1) / (x − 3)

Label the numerator f and denominator g, then apply the formula (f′g − fg′) / g²:

Identify f and g. f = x² + 1, g = x − 3.
Differentiate each. f′ = 2x, g′ = 1.
Apply the formula. (f′g − fg′) / g² = [ 2x(x − 3) − (x² + 1)(1) ] / (x − 3)².
Expand the numerator. 2x² − 6x − x² − 1 = x² − 6x − 1.
Final answer. f′(x) = (x² − 6x − 1) / (x − 3)².

Common sign error: In the numerator of the quotient rule, the second term is subtracted: f′g minus fg′. Students frequently add instead of subtract, changing the sign of the entire numerator.

Common Mistakes to Avoid

Mistake 1 — Forgetting the chain rule inside trig functionsd/dx(sin(2x)) = cos(2x) · 2, not just cos(2x). Whenever the argument is not simply x, the chain rule must be applied.

Mistake 2 — Sign error in the quotient rule numeratorThe formula is (f′g − fg′) / g². The second term is subtracted, not added. Write out both terms before substituting to keep the signs clear.

Mistake 3 — Applying the power rule to eˣThe derivative of eˣ is eˣ — it does not follow the power rule. The power rule applies to xⁿ where the base is the variable. In eˣ, the exponent is the variable.

Mistake 4 — Dropping constants before differentiatingDifferentiate the full expression first, then simplify. Do not remove constant coefficients partway through a product or quotient rule application.

Mistake 5 — Not simplifying after the product or quotient ruleAlways expand and simplify the numerator (or terms) after applying these rules. An unsimplified answer is considered incomplete on most exams.

Frequently Asked Questions

What is the derivative of a constant?

Zero. The derivative of any constant c is 0, because a constant has no rate of change. For example, d/dx(7) = 0 and d/dx(−100) = 0.
What is the derivative of sin(x)?

The derivative of sin(x) is cos(x). This is one of the fundamental trigonometric derivatives. The derivative of cos(x) is −sin(x) — note the negative sign.
How do you find the second derivative?

Differentiate the function once to get f′(x), then differentiate f′(x) again to get f″(x). The second derivative measures how the rate of change itself changes — used in concavity tests and identifying inflection points.
What is the difference between a derivative and an integral?

A derivative measures the instantaneous rate of change of a function (differentiation). An integral accumulates the total area under a curve (integration). They are inverse operations, connected by the Fundamental Theorem of Calculus. Use the Integral Calculator for integration problems.
What does the derivative tell you about a graph?

The derivative at a point equals the slope of the tangent line at that point. Where the derivative is positive, the function is increasing. Where it's negative, the function is decreasing. Where it equals zero, there may be a local maximum or minimum.